Canonical ensemble approximation

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

According to equation (21) the systems of the canonical ensemble are conservative. Each system moves independently of all others and the phase integral exists for each of them. Each system therefore moves on a surface of constant energy and corresponds to a microeanonical ensemble. In this sense the canonical ensemble is built up from a multitude of microeanonical ensembles. Quantities defined for the microeanonical ensemble may therefore be averaged over the canonical ensemble. The original system which is represented by the canonical ensemble however, cannot be described, even approximately, as conservative. It is customary to denote the Hamiltonian of the systems of the canonical ensemble as the Hamiltonian of the original system, which is not justified. 

The pairwise nature of the bond-boost makes this task easier since such traps would show up as a non-convexity of some of the biased effective pair potentials, which in the canonical ensemble can be taken to be the pairwise potential of mean force (PMF, denoted as V). Thus, assuming that V is approximately quadratic for lei safety condition can be enforced by setting Sa[Pg.92]

Rel ation of Exact TST to the Harmonic Approximation. In the canonical ensemble, the most familiar TST expression for the rate constant is probably... 

TST22.23 also makes the statistical approximation and invokes an equihbrium between reactant and TS. TST invokes constant temperature instead of a micro-canonical ensemble as in RRKM theory. Using statistical mechanics, the reaction rate is given by the familiar equation... 

The electronic ground-state energy /i TV, v for a given average number of electrons N and the external potential v of the BO approximation is also the functional of the grand-canonical ensemble density [123] ... 

AN APPROXIMATE GRAND CANONICAL ENSEMBLE THE EQUILIBRIUM ENSEMBLE... 

The one-field model gives a description of the approximate grand canonical ensembles introduced in Chapter 11, Section 5. These ensembles, called equilibrium ensembles , depend on only two fugacities, which determine respectively the average number of polymers and the average number of monomers (constituting the polymers). In the continuous case, a connected partition function J(/, a) can be associated with this ensemble it is defined by... 

Every serious student of fluids will own a copy of Rowlinson s book on liquids and liquid mixtures, and there is no warrant for a repetition of his scholarly and lucid exposition of the classical thermodynamics of the critical state. But a few of the important points must be brought to mind. Consider the classical isotherm portrayed in Fig. 1. The solid fine represents the observed pressure of a system in its most stable state of volume V. Between points 1 and 4 the compressibility of the fluid is infinite, although approximate statistical-mechanical theories, when based on the canonical ensemble, give a loop between points 1 and 4 and... 

Modeling the electrochemical system from first principles presents a considerable challenge. First, the system potential is controlled by combined chemical interactions at the anode and cathode, each with their respective double layer and connected via macroscopic distances of metallic conductor and solution. Such a scale is not accessible to quantum-mechanical calculations. This dilemma is typically resolved, however, by modeling the anode and cathode separately as artificially charged half-cells or approximate models of half-cells. Second, quantum-mechanical calculations are usually performed using a canonical ensemble, in which the number of electrons is the... 

We can, however, begin to model the system instead as an electrochemical half-cell using the canonical ensemble. In order to do so, we must (1) enable the system potential to be extracted from the calculation such that it can be related to a known reference potential and (2) allow the system potential to be accurately controlled in order to access a range of electrochemical conditions. A variety of approximate quantum-mechanical approaches have been developed to extract the influence of the electrochemical potential on the surface chemistry (and reactivity) of adsorbed species. These methods, briefly reviewed below, typically differ from one another in the model of the electrode surface and how the potential is applied to the electrode surface. 

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